Handbook of the History of Logic: - Fordham University Faculty

110 Ian Wilks
to represent the content of that form correctly. 52 But Abelard’s conclusion is not
that there is anything illegitimate about propositions with interposed negation.
His point is simply that if we start with the proposition “All S is P ,” and then
build a square of opposition around it, no proposition in that square will contain a
separative negation. The sort of negation which produces the logical oppositions
is not separative. But this does not mean that separative negations can never
appear in a square of opposition. In fact, to the square beginning with “All S is
P ” Abelard constructs corresponding square beginning with “All S is not P ”; the
corresponding E-form is “No S is not P ,” the I-form is “Some S is not P ”and
the O-form is “Not all S is not P ” [Abelard, 1927, p. 407 (20–31); Abelard, 2006,
07.34–35]. The point is that “All S is not P ” is treated not as negated form of “All
S is P ,” but as a distinct predication, itself subject to placement as the A-entry
on a square of opposition. Given the two different kinds of negation, therefore,
“All S is P ” in fact gives rise to two squares of opposition — or, we might say, a
double square (or, as it has been called, a rectangle of opposition [Martin, 2004a,
p. 168]).
That contrary and contradictory propositions are opposed “not from the nature
of the terms but from the very force of assertion” [Abelard, 1927, p. 410 (3–5);
Abelard, 2006, 07.45] is a basic point about propositional structure which makes
it possible to understand how propositional connectives, like negation, operate. 53
This was later to be a key discovery of Gottlob Frege, and the basis for many of
his innovations. But it is already present in the work of Abelard. 54 It confers
upon him not only a viable basis for constructing a propositional logic, but gives
him heightened awareness of the importance of scope in understanding the syntax
of propositions under logical analysis — in this case the scope of negation. This
awareness naturally shapes his account of modal theory as well, where issues of
scope are so ubiquitous. In modal theory, however, his preferences attach to a
narrow scope of logical operation, not a wide one, so that he privileges modal
propositions formulated with interposed modal operators (the syntactic counterparts
of separative negations).
When interposed, modal operators take the form of adverbs. A mode in effect
answers a “how” question; one asks “How did he read?” and the respondent
formulates the answer by modifying the verb with a word like “well,” “badly” or
“quickly” [Abelard, 1958, p. 3 (16–21); Abelard, 2006, 12.3]. All three adverbs
52 Note that the problem of interposed negation is less likely to arise for the E-proposition,
since it is normally expressed with preceding a sign of negation. The standard Latin quantifying
term for the E-proposition is nullus, which is an etymological product of non and ullus, andin
effect means “not some” (more literally: “not any”). Nullus already does, in effect, pre-position
the non [Abelard, 1970, p. 177 (33–34)].
53 As we shall see below, Abelard realizes the implications of this analysis for understanding
the proper negated form of a conditional [Abelard, 1927, p. 406 (29–31); Abelard, 2006, 07.28].
54 Peter Geach refers to this insight as the “Frege point” [Geach, 1965, p. 449]. Christopher
Martin, who cites Geach on this point [Martin, 1991, p. 281], has stressed the significance of
finding the Frege point already fully articulated by Abelard: “If he was the first to achieve
this understanding in the Middle Ages, and there is no evidence to the contrary, he must be
recognized as one of the greatest of all philosophical logicians” [Martin, 2004a, p. 166].